Unique Minimum Research Articles

Money as minimal complexity

We consider mechanisms that provide the opportunity to exchange commodity i for commodity j, for certain ordered pairs ij. Given any connected graph G of opportunities, we show that there is a unique “G-mechanism” that satisfies some natural conditions of “fairness” and “convenience”. Next we define time and price complexity for any G-mechanism as (respectively) the time required to exchange i for j, and the information needed to determine the exchange ratio (each for the worst pair ij). If the number of commodities exceeds three, there are precisely three minimally complex G-mechanisms, where G corresponds to the star, cycle and complete graphs. The star mechanism has a distinguished commodity — the money — that serves as the sole medium of exchange and mediates trade between decentralized markets for the other commodities. Furthermore, for any weighted sum of complexities, the star mechanism is the unique minimizer of the sum for large enough m.

A quantitative form of Faber–Krahn inequality

The Faber–Krahn inequality states that balls are the unique minimizers of the first eigenvalue of the p-Laplacian among all sets with fixed volume. In this paper we prove a sharp quantitative form of this inequality. This extends to the case $$p>1$$ a recent result proved by Brasco et al. (Duke Math J 164:1777–1831, 2015) for the Laplacian.

A note on Willmore minimizing Klein bottles in Euclidean space

We show that φ∘τ˜3,1:K→R4×{0}n−4 is the unique minimizer among immersed Klein bottles in its conformal class, where φ:S4→R4 is a stereographic projection and τ˜3,1 is the bipolar surface of Lawson's τ3,1-surface [11]. We conjecture that φ∘τ˜3,1 is the unique minimizer among immersed Klein bottles into Rn, n≥4, whose existence the authors and P. Breuning proved in [2].

Tiling the Boolean lattice with copies of a poset

Let P be a partially ordered set with a unique maximal and minimal element, and size 2m, where m is a positive integer. Settling a conjecture of Lonc, we prove that if n is sufficiently large, then the Boolean lattice 2[n] can be partitioned into isomorphic copies of P. Also, we show that if P has a unique maximum and minimum, but the size of P not necessarily a power of 2, then there exists a constant c = c(P) such that all but at most c elements of 2[n] can be covered by disjoint copies of P.

Adaptive Source Localization Based Station Keeping of Autonomous Vehicles

We study the problem of driving a mobile sensory agent to a target whose location is specified only in terms of the distances to a set of sensor stations or beacons. The beacon positions are unknown, but the agent can continuously measure its distances to them as well as its own position. This problem has two particular applications: (1) capturing a target signal source whose distances to the beacons are measured by these beacons and broadcasted to a surveillance agent, (2) merging a single agent to an autonomous multi-agent system so that the new agent is positioned at desired distances from the existing agents. The problem is solved using an adaptive control framework integrating a parameter estimator producing beacon location estimates, and an adaptive motion control law fed by these estimates to steer the agent toward the target. For location estimation, a least-squares adaptive law is used. The motion control law aims to minimize a convex cost function with unique minimizer at the target location, and is further augmented for persistence of excitation. Stability and convergence analysis is provided, as well as simulation results demonstrating performance and transient behavior.

Positive loops and $$L^{\infty }$$ L ∞ -contact systolic inequalities

We prove an inequality between the $L^{\infty}$-norm of the contact Hamiltonian of a positive loop of contactomorphims and the minimal Reeb period. This implies that there are no small positive loops on hypertight or Liouville fillable contact manifolds. Non-existence of small positive loops for overtwisted 3-manifolds was proved by Casals-Presas-Sandon in [CPS16]. As corollaries of the inequality we deduce various results. E.g. we prove that certain periodic Reeb flows are the unique minimizers of the $L^\infty$-norm. Moreover, we establish $L^\infty$-type contact systolic inequalities in the presence of a positive loop.

A natural approach to the asymptotic mean value property for the p-Laplacian

Let $1\le p\le\infty$. We show that a function $u\in C(\mathbb R^N)$ is a viscosity solution to the normalized $p$-Laplace equation $\Delta_p^n u(x)=0$ if and only if the asymptotic formula $$ u(x)=\mu_p(\ve,u)(x)+o(\ve^2) $$ holds as $\ve\to 0$ in the viscosity sense. Here, $\mu_p(\ve,u)(x)$ is the $p$-mean value of $u$ on $B_\ve(x)$ characterized as a unique minimizer of $$ \inf_{\la\in\RR}\nr u-\la\nr_{L^p(B_\ve(x))}. $$ This kind of asymptotic mean value property (AMVP) extends to the case $p=1$ previous (AMVP)'s obtained when $\mu_p(\ve,u)(x)$ is replaced by other kinds of mean values. The natural definition of $\mu_p(\ve,u)(x)$ makes sure that this is a monotonic and continuous (in the appropriate topology) functional of $u$. These two properties help to establish a fairly general proof of (AMVP), that can also be extended to the (normalized) parabolic $p$-Laplace equation.

Hyperbolicity of Minimizers and Regularity of Viscosity Solutions for a Random Hamilton–Jacobi Equation

We show that for a large class of randomly kicked Hamilton–Jacobi equations, the unique global minimizer is almost surely hyperbolic. Furthermore, we prove that the unique forward and backward viscosity solutions, though in general only Lipshitz, are smooth in a neighborhood of the global minimizer. Related results in the one-dimensional case were obtained by E, Khanin et al. (Ann Math (2) 151(3):877–960, 2000). However, the methods in the above paper are purely one-dimensional and cannot be extended to the case of higher dimensions. Here we develop a completely different approach.

Plateau-Rayleigh instability in solids is a simple phase separation.

A long elastic cylinder, with radius a and shear-modulus μ, becomes unstable given sufficient surface tension γ. We show this instability can be simply understood by considering the energy, E(λ), of such a cylinder subject to a homogenous longitudinal stretch λ. Although E(λ) has a unique minimum, if surface tension is sufficient [Γ≡γ/(aμ)>sqrt[32]] it loses convexity in a finite region. We use a Maxwell construction to show that, if stretched into this region, the cylinder will phase-separate into two segments with different stretches λ_{1} and λ_{2}. Our model thus explains why the instability has infinite wavelength and allows us to calculate the instability's subcritical hysteresis loop (as a function of imposed stretch), showing that instability proceeds with constant amplitude and at constant (positive) tension as the cylinder is stretched between λ_{1} and λ_{2}. We use full nonlinear finite-element calculations to verify these predictions and to characterize the interface between the two phases. Near Γ=sqrt[32] the length of such an interface diverges, introducing a new length scale and allowing us to construct a one-dimensional effective theory. This treatment yields an analytic expression for the interface itself, revealing that its characteristic length grows as l_{wall}∼a/sqrt[Γ-sqrt[32]].

English

English

Variational formulas and disorder regimes of random walks in random potentials

We give two variational formulas (qVar1) and (qVar2) for the quenched free energy of a random walk in random potential (RWRP) when (i) the underlying walk is directed or undirected, (ii) the environment is stationary and ergodic, and (iii) the potential is allowed to depend on the next step of the walk which covers random walk in random environment (RWRE). In the directed i.i.d. case, we also give two variational formulas (aVar1) and (aVar2) for the annealed free energy of RWRP. These four formulas are the same except that they involve infima over different sets, and the first two are modified versions of a previously known variational formula (qVar0) for which we provide a short alternative proof. Then, we show that (qVar0) always has a minimizer, (aVar2) never has any minimizers unless the RWRP is an RWRE, and (aVar1) has a minimizer if and only if the RWRP is in the weak disorder regime. In the latter case, the minimizer of (aVar1) is unique and it is also the unique minimizer of (qVar1), but (qVar2) has no minimizers except for RWRE. In the case of strong disorder, we give a sufficient condition for the nonexistence of minimizers of (qVar1) and (qVar2) which is satisfied for the log-gamma directed polymer with a sufficiently small parameter. We end with a conjecture which implies that (qVar1) and (qVar2) have no minimizers under very strong disorder.

Relating Mass to Angular Momentum and Charge in Five-Dimensional Minimal Supergravity

We prove a mass-angular momentum-charge inequality for a broad class of maximal, asymptotically flat, bi-axisymmetric initial data within the context of five-dimensional minimal supergravity. We further show that the charged Myers-Perry black hole initial data are the unique minimizers. In addition, we establish a rigidity statement for the relevant BPS bound, and give a variational characterization of BMPV black holes.

Image Regularity and Fidelity Measure with a Two-Modality Potential Function

We define a strictly convex smooth potential function and use it to measure the data fidelity as well as the regularity for image denoising and cartoon-texture decomposition. The new model has several advantages over the well-known ROF or TV-L2 and the TV-L1 model. First, due to the two-modality property of the new potential function, the new regularity has strong regularizing properties in all directions and thus encourages removing noise in smooth areas, while, near edges, it smoothes the edge mainly along the tangent direction and thus can well preserve the edges. Second, the new potential function is very close to the L1 norm; thus using it to measure the data fidelity makes the new model perform very well in removing impulse noise and preserving the contrast. Lastly, the proposed fidelity and regularization term is strictly convex and smooth and thus allows a unique global minimizer and it can be solved by using the steepest descent method. Numerical experiments show that the proposed model outperforms TV-L2 and TV-L1 in removing impulse noise and mixed noise. It also outperforms some state-of-the-art methods specially designed for impulse noise. Tests on cartoon-texture decomposition show that our method is effective and performs better than TV-L1.

The $R_{\mathrm{\scriptscriptstyle ON},\mathrm {min}}$ of Balanced Symmetric Vertical Super Junction Based on R-Well Model

The global optimization of the balanced symmetric vertical superjunction (VSJ) device is proposed based on the R-well model for the first time to realize the unique minimum specific on-resistance $R_{{\mathrm{\scriptscriptstyle ON},\mathrm {min}}}$ . The R-well model, which is originated from our previous VSJ mode theory, shows the relationship between $R_{{\mathrm{\scriptscriptstyle ON}}}$ and the doping concentration N under the given pillar width W and breakdown voltage $V_{B}$ . The global $R_{{\mathrm{\scriptscriptstyle ON}}}$ optimization is realized to obtain the design formulas of N and the pillar length $L_{d}$ . The calculated results are in good agreement with the simulations. It is demonstrated from the comparisons with the simulations and the existing experiments that the optimization in this paper realizes the unique $R_{{on,\mathrm {min}}}$ with a relationship of $R_{{\mathrm{\scriptscriptstyle ON}}}\propto V_{B}^{1.03}$ . The R-well model is also universal for other $R_{on}$ optimizations.

Approximation of constrained problems using the PGD method with application to pure Neumann problems

In this paper we introduce, analyze, and compare several approaches designed to incorporate a linear (or affine) constraint within the Proper Generalized Decomposition framework. We apply the considered methods and numerical strategies to two classes of problems: the pure Neumann case where the role of the constraint is to recover unicity of the solution; and the Robin case, where the constraint forces the solution to move away from the already existing unique global minimizer of the energy functional.

A spectral isoperimetric inequality for cones

In this note we investigate three-dimensional Schr\"odinger operators with $\delta$-interactions supported on $C^2$-smooth cones, both finite and infinite. Our main results concern a Faber-Krahn-type inequality for the principal eigenvalue of these operators. The proofs rely on the Birman-Schwinger principle and on the fact that circles are unique minimizers for a class of energy functionals. The main novel idea consists in the way of constructing test functions for the Birman-Schwinger principle.

Magnetic wells in dimension three

This paper deals with semiclassical asymptotics of the three-dimensional magnetic Laplacian in presence of magnetic confinement. Using generic assumptions on the geometry of the confinement, we exhibit three semiclassical scales and their corresponding effective quantum Hamiltonians, by means of three microlocal normal forms \`a la Birkhoff. As a consequence, when the magnetic field admits a unique and non degenerate minimum, we are able to reduce the spectral analysis of the low-lying eigenvalues to a one-dimensional $\hbar$-pseudo-differential operator whose Weyl's symbol admits an asymptotic expansion in powers of $\hbar^{\frac1 2}$.

NOTE ON THE SMITH–WILSON INTEREST RATE CURVE

Since the entry into force of Solvency II as of 1 January 2016, all European insurance companies concerned have to use the Smith–Wilson interest rate curve to determine the value of their insurance obligations and thus of a substantial part of their balance sheet. Although Smith & Wilson introduce the underlying discount curve [Formula: see text] as the sum of a ‘long-term’ discount curve [Formula: see text] and a linear combination of the so-called Wilson function [Formula: see text] evaluated at different payment dates [Formula: see text], that is, [Formula: see text] a mathematically sound derivation of its shape is missing in the literature. The aim of this paper is to close this gap. To this end, we reformulate the infinite-dimensional optimization problem stated in Smith & Wilson (2000) within an analytically rigorous framework. We prove that it has a unique minimizer and explicitly derive the formula displayed above. In doing so, we show that [Formula: see text] is in fact the kernel function of a particular reproducing kernel Hilbert space, which is the key result to fully understanding the shape of [Formula: see text].

A uniqueness result for a class of non-strictly convex variational problems

Let Ω be a smooth domain in R2, we prove that if g:[0,+∞)→[0,+∞] is convex with g(0)<g(t) whenever t>0 then there exists an unique minimizer u∈C0,1(Ω) of the functional u↦∫Ωg(|∇u|)dxdy among all Lipschitz-continuous functions that assume the same value of u on ∂Ω.

Shell completion of helium atoms around the coronene cation

Recent mass spectrometry measurements carried out for the coronene (C24H12) cation coated with helium atoms have revealed special stabilities associated with ’magic numbers’ at sizes 38, 41, and 44, the latter being interpreted as the signature of a complete first solvation shell. In the present contribution we examine by means of path-integral molecular dynamics simulations employing a polarizable potential the equilibrium structures of such clusters and how they differ from the classical global minima. Our results confirm the special energetic stability of the suggested sizes and the highly symmetric geometric arrangement of the helium atoms around the aromatic cation. However, they also indicate that the helium atoms in the vicinity of the aromatic plane are highly fluxional and not associated with a unique local minimum in the energy landscape. Above the size of 44, the shell expands but remains stable with no immediate evidence for a second shell.